Sets of Formulas Valid in Finite Structures

Alan L. Selman
1973 Transactions of the American Mathematical Society  
line ^4 d-{V ) « d(V ) vdfVJ, SETS OF FORMULAS VALID IN FINITE STRUCTURES Abstract A function Ir is defined on the set of all subsets of u) so that for each set K, the value, Ir , is the set of formulas K valid in all structures of cardinality in K. An analysis is made of the dependence of \s on K. It is easily seen that K for all infinite sets K, d(K) V 1 £ d(K) £ d(K) ! . On the other hand, we prove that d(U ) = d(lr ) = d(U ) , and use this tõ KVj ^ K ~ u prove that for any two degrees a and
more » ... b, a ^ 1, a cp £ ID . K. K K K to and rn defined above have conceptual interest, and, by Lemma 1, for each set K, d(V ) = d(ft\__) = d(to ) . In fact, we prefer to analyze the function to, since as is easily seen, for all K, to is r.e. in K.
doi:10.2307/1996611 fatcat:rac7aqrqxvgpfegy6peul43xem